The issue you describe is more often likely going to be a connection issue. Now you can get disconnected or stalled on Overwatch due to a connection issue, but your general internet connection is not affected. This is because there is something between you and the game server on the connection route that can be causing the issue. It can be frustrating to pinpoint when it happens where you can’t even load into the game. The best advice is to try and run a WinMTR test which pings each step of your connection route to the game servers. Remember to try and run this test during times you are seeing this issue. If you need help reviewing the results please post them in the technical support forum as Blizzard Technical Support will monitor that forum (not here in the Bug Report forum).
Unfortunately once you are selected for a match, you are held accountable for it. Penalties are still applied even if you can’t get in (this is to prevent queue dodging.) You can learn about Blizzard’s policy about this here: